Harmonic analysis on Euclidean spaces Convergence of Fourier series Theorem

Hunt's extension of Carleson's theorem

Builds on Carleson's theorem Lp space Fourier series Fourier coefficient Almost everywhere

Statement

Let $1 < p < \infty$ . Let $f \colon \R \to \C$ be a $2\pi$ -periodic function with $\int_{-\pi}^{\pi} \abs{f}^p < \infty$ , that is, $f \in L^p(\mathbb{T})$ (see Lp space). Let $S_N f(x) = \sum_{k=-N}^{N} \hat f(k) e^{ikx}$ be the partial sums of its Fourier series, where $\hat f(k)$ are its Fourier coefficients. Then $S_N f(x) \to f(x)$ as $N \to \infty$ for almost every $x \in \R$ .

Remarks

Hunt proved this in 1968 by extending the method of Carleson's theorem, which is the case $p = 2$ . The key estimate is that the maximal operator $f \mapsto \sup_N \abs{S_N f}$ is bounded on $L^p(\mathbb{T})$ for $1 < p < \infty$ . The theorem is new only for $1 < p < 2$ , since $L^p(\mathbb{T}) \subset L^2(\mathbb{T})$ for $p \ge 2$ . The case $p = 1$ fails: Kolmogorov's example is an integrable function whose Fourier series diverges everywhere.

Going deeper

For a student

The proof shows that the Carleson maximal operator is bounded on $L^p$ for $1 < p < \infty$ . With the maximal bound, a.e. convergence follows from convergence on the dense class of trigonometric polynomials. The exact borderline between $L^1$ and the spaces $L^p$ with $p > 1$ for almost everywhere convergence is still studied.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Almost everywhere concept
Fourier coefficient concept
Fourier series concept
Lp space concept
Carleson's theorem statement
Hunt's extension of Carleson's theorem statement

Tag	`NAX51M6025`
URI	`m2://NAX51M6025`
Kind	statement, theorem
Writing	W0 stub
MSC	42A20, 42B25
Status	This copy of the site has no build data. Run `m2 status NAX51M6025`.
About	Fourier series, Convergence of Fourier series
Uses	Carleson's theorem, Lp space, Fourier series, Fourier coefficient, Almost everywhere